small enough, say smaller than ε∕2, the integrand is no larger than Cte−εte

ε
2

t = Cte−

ε
2

t which is
in L1 and so the dominated convergence theorem applies and it follows that f′

(s)

= ∫0∞

(− t)

e−stϕ

(t)

dt
whenever Re

(s)

> λ. In fact all the derivatives will exist, by the same argument, but this will
follow more easily as a special case of more general results when we get around to using this.
■

The whole approach for Laplace transforms in differential equations is based on the assertion that if
ℒ

(f)

= ℒ

(g)

, then f = g. However, this is not even true because if you change the function on a
set of measure zero, you don’t change the transform. However, if f,g are continuous, then it
will be true. Actually, it is shown here that if ℒ

(f)

= 0, and f is continuous, then f = 0.
The approach here is based on the Weierstrass approximation theorem or rather a case of
it.

With the Lebesgue fundamental theorem of calculus which has not been presented, the above argument
will show that it is only necessary to assume the functions are in L1 and conclude they are equal a.e. if
their Laplace transforms are equal.